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Computability logic (CoL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is arecently introduced semantical platform and ambitious program for redevelopinglogic as a formal theory of computability, as opposed to the formal theory oftruth that logic has more traditionally been. Its expressions representinteractive computational tasks seen as games played by a machine against theenvironment, and "truth" is understood as existence of an algorithmic winningstrategy. With logical operators standing for operations on games, theformalism of CoL is open-ended, and has already undergone series of extensions.This article extends the expressive power of CoL in a qualitatively new way,generalizing formulas (to which the earlier languages of CoL were limited) tocircuit-style structures termed cirquents. The latter, unlike formulas, areable to account for subgame/subtask sharing between different parts of theoverall game/task. Among the many advantages offered by this ability is that itallows us to capture, refine and generalize the well knownindependence-friendly logic which, after the present leap forward, naturallybecomes a conservative fragment of CoL, just as classical logic had been knownto be a conservative fragment of the formula-based version of CoL. Technically,this paper is self-contained, and can be read without any prior familiaritywith CoL.

From formulas to cirquents in computability logic